name the quadrant in which the angle \\( \\theta \\) lies. \\( \\sin \\theta<0, \\cot \\theta>0 \\) choose…

Explanation:

Step1: Analyze the sign of (\sin\theta)

Recall that (\sin\theta=\frac{y}{r}), where (r = \sqrt{x^{2}+y^{2}}>0). If (\sin\theta<0), then (y < 0). So the angle (\theta) lies in either Quadrant III or Quadrant IV.

Step2: Analyze the sign of (\cot\theta)

Recall that (\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{x}{y}). Since (\cot\theta>0), then (x) and (y) have the same sign. From Step 1, we know (y < 0), so (x<0) as well.

Step3: Determine the quadrant

When (x < 0) and (y < 0), the angle (\theta) lies in Quadrant III.

Answer:

Quadrant III